What Special About This Number
red for the properties specially in dozenal (dependent in which base is used, in this wiki, we always use the dozenal base). 0 to 100 0 is the additive identity. 1 is the multiplicative identity. 2 is the only even prime. 3 is the smallest k'' such that it is impossible to construction of an angle equal to 1/''k of a given arbitrary angle using only an unmarked straightedge and a compass. 4 is the smallest number of colors sufficient to color all planar maps. 5 is the smallest k'' such that general algebraic equation with degree ''k cannot be solved algebraically. 6 is the smallest possible order of nonabelian group. 7 is the smallest k'' such that regular ''k-gon is not constructible using a compass and an unmarked straightedge. 8 is the smallest positive integer with no primitive roots. 9 is the only number k'' such that (''k times a triangular number) plus 1 (i.e. centered k''-gonal number) is always also a triangular number. (thus, all generalized repunits base 9 are also triangular numbers, and 9 is the only such base) X is the smallest noncototient number. E is the largest squarefree number ''n such that the quadratic field OQ(√−n) is a Euclidean domain. 10 appears in the value of the Riemann zeta function at −1 (i.e. ζ(−1) = −1/10, which equals the negative value of the reciprocal of 10, or the additive inverse of the multiplicative inverse of 10). 11 is the number of Archimedean solids. 12 is the smallest nontotient number. 13 is the smallest k''>1 such that the number of terms of the ''k-th cyclotomic polynomial does not equal to the largest prime factor of k''. 14 is the only number of the form ''ab'' = ''ba'', with ''a, b'' nonnegative integers, ''a ≠ b''. 15 is the only positive Genocchi prime. (if we allow negative primes, then −3 is the only other such prime) 16 is the smallest proven solitary number which is not coprime to its sum-of-divisors. 17 is the smallest number (and the only number ≤50) not appearing in the first 100 terms of Recamán's sequence (in fact, 17 does not appear in the first 49800 terms of Recamán's sequence, the first time 17 appearing is term 49872, if 0 is term 0, 1 is term 1). 18 is the number of moves (quarter or half turns) required to optimally solve a Rubik's Cube in the worst case. 19 is the smallest number of distinct squares needed to tile a square. 1X is the numerator of an approximation of π (1X/7). 1E is the smallest number ''n such that the relative class number h− of cyclotomic field Q(e''2πi/''n) is greater than 1. 20 is the largest number for which the Dirichlet characters are all real. 21 is the smallest square that can be written as a sum of 2 positive squares. 22 is the only positive number to be directly between a square and a cube. 23 is the number n'' for which (the largest number in the 3''x+1 sequence starting at n'')/(''n''2) is largest. (i.e. 5414/(23^2) = 10.7E7314) 24 is the smallest even number which is a (Fermat) pseudoprime to some nontrivial bases. 25 is the largest number ''n such that 2''x''2 + n'' is prime for all 0≤''x≤''n''−1. (since it is divisible by n'' for ''x = n'', one cannot do be better than this) 26 is the largest number with the property that all smaller numbers relatively prime to it are prime or 1. 27 is the smallest number ''n such that integral0∞(cos(x'')cos(''x/2)cos(x''/3)...cos(''x/''n'')) ≠ π/2. 28 is the smallest number n'' such that the ''n-th row of the modulo-2 Pascal's triangle (the top row, which contains only one 1, is the 0th row, not the 1st row), when read in binary, is not a number of the sides of constructible regular polygon. 29 is the largest number that is not a sum of distinct triangular numbers. 2X is the smallest number with the property that it and its neighbors have the same number of divisors. 2E is the smallest semiprime which cannot be primary pretender. (note that all primary pretenders except the largest (3X9, which is the smallest Carmichael number) are semiprimes) 30 is the smallest perfect power which is not a prime power. 31 is the smallest irregular prime. 32 is the magic constant of the only non-trivial normal magic hexagon. 33 is the smallest n'' which is not power of 10 such that ''n.n''.''n...n''.''n.1 (dot means concatenation) cannot be prime. 34 is the smallest n'' such that ''n.111...111 (dot means concatenation) cannot be prime. 35 is the largest number n'' such that ''x''2 + ''x + n'' is prime for all 0≤''x≤''n''−2. (since it is divisible by n'' for ''x = n''−1, one cannot do be better than this) 36 is the largest number of sides of a regular polygon that can fill a point with other regular polygons. 37 is the smallest number ''n such that (define a''n'': a''0 = 1, for ''k > 0, a''k'' = (1+''a''02+''a''12+...+''a'k''−12)/''k'') ''a'n'' is not integer. (a''n'' is the Göbel's sequence) (the fractional part of a''37 is 20/37) 38 is the smallest ''n such that all of n''.0, ''n.1, n''.2, ..., ''n.E (dot means concatenation) are composite. (i.e. all of 10''n''+0, 10''n''+1, 10''n''+2, ..., 10''n''+E are composite) 39 is the smallest odd positive integer that is not power of squarefree number. 3X is the largest even number which is a value of n'' for incrementally largest values of minimal ''x satisfying Pell equation x^2−Dy^2=1. 3E is the smallest base for which no generalized Wieferich primes are known. 40 is the largest number n'' such that the sum of the first ''n positive triangular numbers is also a triangular number. 41 is the smallest number with the property that it and its neighbors are not squarefree. 42 is the smallest number which can be written as the sum of of 2 positive squares in 2 different ways. 43 is the number of groups with order 25 (=28). 44 is the smallest untouchable number > 5 (the conjectured only odd untouchable number). 45 is the smallest prime that produces prime reciprocal magic square. 46 is the smallest totient number which is not totient of squarefree number. 47 is the largest triangular number in the Fibonacci sequence. 48 is the only number n such that no x^2 mod n is prime and n is not Euler's "numerus idoneus" (or convenient numbers, or idoneal numbers). 49 is the smallest number >1 of the form Φ''n''(2) which is neither prime nor Fermat pseudoprime base 2. 4X is the largest squarefree even number n'' such that the imaginary quadratic field Q(√−n) has class number 2. 4E is the smallest prime factor of the smallest composite Euclid number (i.e. 4E|(11#+1) = 15467 = 4E×365). 50 is the smallest possible order of nonsolvable group. 51 is conjectured to be the largest number ''n such that kn−1 and kn+1 are not both primes for all k'' ≤ 4''n. 52 is the smallest number that can be written as the sum of of 3 distinct squares in 2 ways. 53 is the largest number of the form a''n'' − b''n'' with no primitive prime factors (26 − 16). 54 is the smallest number >1 which is both square number and cube number. 55 is the smallest deceptive prime. 56 is the denominator of the first Bernoulli number whose absolute value is not a unit fraction (B''X = 5/56). 57 is the smallest prime which is both Bernoulli irregular and Euler irregular. 58 is the smallest ''n which is not power of 10 and not congruent to 1 mod 11 (in which all such numbers are divisible by 11) such that n''k''.1 (dot means concatenation) is composite for all 1≤''k''≤1000. (the smallest k''≥1 such that this number is prime is 2781E5) 59 is the number in the square root of the algebraic form of the plastic number. 5X is the smallest weird number. 5E is the largest number whose square is one more than a factorial number. 60 is the smallest Achilles number. 61 is the largest squarefree number ''n such that the quadratic field OQ(√n) is a Euclidean domain. 62 is the number of different non-Hamiltonian polyhedra with a minimum number of vertices. 63 is the number of uniform polyhedra, excluding the infinite sets. 64 is the smallest n'' such that ''n-Fibonacci numbers cannot be primes. 65 is the largest number that cannot be written as a sum of distinct numbers whose reciprocals sum to 1. 66 is the dimension of the exceptional Lie group E''6. 67 is the smallest prime number ''p for which the real quadratic field Q√p has class number greater than 1. 68 is the number of single-tile moves required to optimally solve a Fifteen game in the worst case. 69 is the only known square n'' such that ''n×2''n''−1 is prime (Woodall prime). (note that its square root is also a square) 6X is the number of 6-hexes. 6E is the smallest odd prime p'' (let a(''p) is the smallest generalized Wieferich prime base p'') such that a(a(''p)) = p''. 70 is the smallest number ''n such that n'' is neither squarefree nor of the form ''pa''q'b'' with p'', ''q primes, but no simple group with order n'' exists. 71 is the largest number ''n such that the sum of the first n'' positive square numbers is a triangular number. 72 has a 4th root that starts 3.0662666762266... (there are 8 6's in the first 11 digits after the dozenal point) 73 is the sum of the squares of the first four primes. 74 is the smallest abundant number coprime to the smallest odd abundant number (669). 75 is the smallest prime to start a Cunningham chain of the first kind of ≥6 terms. (note that 2 starts a Cunningham chain of the first kind of 5 terms) 76 is the smallest pronic number which is nontotient. (note that pronic number n×(n−1) cannot be nontotient if p is prime, since this number equals φ(p2) 77 is the smallest positive integer expressible as a sum of two cubes in two different ways if negative roots are allowed. 78 is the largest possible number of faces of an Archimedean solid. 79 is the smallest number ''n such that the distance from n'' to closest prime is >3. 7X is the smallest number ''n>1 such that M''(''n) is positive, where M'' is the Mertens function. 7E is the third-smallest number whose aliquot sequence terminates at 6 (within the sequence {7E, 21, 6}). 80 is the only number ''n besides 2 with the property that not n'' but ''n/2 is a value of k'' for incrementally largest values of groups of order ''k sets a record. 81 is the smallest base not of the form n''2''k+1 or of the form 4''n''4 (where generalized Wagstaff numbers can be factored algebraically) for which no generalized Wagstaff (probable) primes are known. 82 is the only known number n'' of the form 2''p''2 with odd prime ''p such that Φ''n''(2) is (probable) prime. 83 is the smallest number whose factorial is greater than googol (=10100). 84 is the smallest number with more than one factorization into L''-primes. (Let ''L = {1, 4, 7, X, 11, 14, 17, 1X, ..., 3''k''+1, ...}; then an L''-prime is in ''L but is not divisible by any members of L'' except itself and 1. 85 is the alternating factorial of 5. 86 is the smallest number ''n for which (let n'' = 2a0 × 3a1 × 5a2 × 7a3 × Ea4 × ...) a0 + a1''x + a2''x''2 + a3''x''3 + ... = 0 does not have algebraic solution (i.e. there is no solution in radicals). 87 is the smallest Lucas-Wieferich prime associated with the pair (P'', ''Q) = (4, 1). 88 is the smallest number of unit line segments that can exist in a plane with four of them touching at every vertex. 89 is the smallest integer such that the factorization of ''x'n''−1 over Q'' includes coefficients other than ±1. In other words, the 89th cyclotomic polynomial, Φ89, is the first with coefficients other than ±1. 8X is the denominator of an approximation of π (239/8X). 8E is the smallest prime ''p ends with E such that 2''p''−1 is prime. 90 is the number of heptominoes (7-minoes). 91 is the number of different families of subsets of a three-element set whose union includes all three elements. 92 is the smallest number n'' such that there are no known powerful number ''k such that k''+''n is also powerful. 93 is the magic constant of the smallest magic square using only 1 and prime numbers. 94 is the side of the smallest square that can be tiled with distinct integer-sided squares. 95 is the denominator of an approximation of π (257/95). 96 is the starting number of the first run of 11 consecutive composite numbers. 97 is the number of rooted trees with 8 vertices. 98 is the final population for the Conway's game of Life starting with the "F-pentomino". 99 is the smallest possible length of the longest side of a Heronian tetrahedron (one whose sides are all rational numbers). 9X is the smallest n'' such that the range ''n, n'' + 1, ... 4''n/3 contains at least one prime from each of these forms: 4''k'' + 1, 4''k'' - 1, 6''k'' + 1 and 6''k'' - 1. 9E is the largest number n'' such that the ''n''th triangular number is also a tetrahedral number. X0 is the smallest number to appear 6 times in Pascal's triangle. X1 is the only Brazilian number with exactly 3 divisors. X2 is the number of partitions of 20 into distinct parts. X3 is the smallest ''k for which there is no known prime of the form (k''−1)×''kn''+1. X4 is the smallest nontotient which is also an untouchable number. X5 is the largest two-digit narcissistic number. X6 is the number of different semigroups on 4 elements (up to isomorphism and reversal). X7 is the smallest de Polignac number. X8 is the largest number that is not a sum of distinct square numbers. X9 is the smallest number that can be written as the sum of 3 squares in 4 ways. XX is the only integer that is the sum of the squares of its first four divisors. XE is the smallest Sophie Germain prime congruent to 3 mod 4 which is not safe prime. E0 is the smallest non-squarefree Catalan number. E1 is the smallest overpseudoprime base 10. E2 is the smallest number whose aliquot sum is a weird number. E3 is the product of the first two odd composite numbers. E4 is the only number which is a self-descriptive number in some base (base 4) which has a smaller self-descriptive number (84). E5 is the smallest strictly non-palindromic number ''n>4 such that n''+2 is also strictly non-palindromic. E6 is the smallest number whose aliquot sequence has length >20 (in fact, >120, its length is 12X). E7 is the largest prime factor among the smallest pair of odd amicable numbers. E8 is the largest number whose square is a tetrahedral number. E9 is the smallest ''n>1 such that n''×2''n+1 is prime (Cullen prime). EX is the number of planar graphs with 6 unlabeled vertices. EE is the only number which is product of twin primes but not brilliant number. 100 is the largest square number in the Fibonacci sequence. 101 to 1000 (selected) 101 is the smallest base for which no generalized Woodall primes are known. 102 is the smallest untouchable number which is a semiprime. 103 is the number of sided 6-hexes. 104 is the largest number n'' such that the primitive part of 2''n+1 was once the largest known prime. (start with 2375−1, the largest known prime is almost always Mersenne prime, i.e. of the form 2''n''−1 (not +1), the only one exception is 16X739×2X5141−1) 105 is the smallest number which is not sum of two prime powers (including 1). 106 is the largest gap between consecutive twin prime pairs less than 1000 (it occurs between {46E, 471} and {575, 577}). 107 is the number of species in the Pokémon Go game. 108 was once the smallest base not of the form n^x (where generalized repunits can be factored algebraically) for which no generalized repunit (probable) primes are known (currently, the smallest such base is 135). (recently, the probable prime (108^110461−1)/107 was found) 109 is the sum of the first 5 positive factorials. 10X is the number of regions in regular nonagon (9-gon) with all diagonals drawn. 10E is the sum of the primes between its smallest and largest prime factor. 110 is the smallest number that is the product of two different substrings. 111 is the smallest irregular prime with irregular index greater than 1. 114 is the smallest number n'' with exactly 10 solutions to the equation φ(''x) = n''. 115 is the smallest Harshad number >10 divisible by neither E nor 10. 116 is the smallest number >100 with terminate reciprocal. 117 is the largest Heegner Number. 119 is the midpoint of the ''n''th larger prime and ''n''th smaller prime for all 1≤''n≤6. 11E is the only prime requiring exactly 8 cubes to express it. 120 is the smallest possible order of noncyclic simple group other than groups of the form A''k'' (which is always a noncyclic simple group for k''≥5). 121 is the largest square number in the Pell sequence. 122 is the smallest number n for which φ(n) and σ(n) are both square. 123 is the only Smarandache number which is also triangular number. 125 is the smallest 3-digit Keith number. 129 is the magic constant of the smallest magic square using only prime numbers. 12E is the smallest nonpalindromic number whose square is palindromic. 130 is the largest possible number of edges of an Archimedean solid. 131 is the largest value ''x satisfying the Ramanujan–Nagell equation. 133 is the largest number that equal the sum of the squares of the digits of their own square. 135 is the smallest composite primeval number. 13E is the smallest prime congruent to 1 mod 17 (this is the case which (let a''(''n) is the smallest k'' such that ''kn+1 is prime) log''n''(a''(''n)−1) is largest (i.e. log17(9) = 0.8E55967E072E...)). (note that 17 is the smallest primitive root mod 13E, this is the second-largest case which smallest primitive root mod p'' (with ''p prime) is larger than √''p'', the largest case is the smallest primitive root mod 2X1 is 19) 140 is the smallest multiple of 10 which is not Harshad number. 141 is the largest number that can be written as ab + ac + bc with 0 < a'' < ''b < c'' in a unique way. 143 is the smallest ''n such that binomial(2''n'', n'') is divisible by ''n''2. 145 is the smallest prime ''p such that none of 2''p''+1, 4''p''+1, 8''p''+1, X''p''+1, 12''p''+1, and 14''p''+1 is prime. (Sophie Germain proved that Fermat's last theorem is true for all odd primes p'' such that at least one of 2''p+1, 4''p''+1, 8''p''+1, X''p''+1, 12''p''+1, and 14''p''+1 is prime) 147 is the largest k'' such that all positive values of ''k−2''n''2 are primes or 1. 148 is the largest number n'' ≤ 100000000 such that |''M(n'')| ≥ (√''n)/2, where M'' is the Mertens function. (Mertens conjectured that |''M(n'')| < √''n for all n'' > 1, this is now known to be false) 14X is the number of distinct (non-isomorphic) directed graphs on four unlabeled vertices, not having any isolated vertices. 14E is the number of integer squares (not necessarily of unit size) can be found in a staircase-shaped polyomino formed by stacks of unit squares of heights ranging from 1 to 10. 150 is the smallest non-semiprime whose square is a triangular number. 151 is the smallest odd number ''D with no prime factors p'' = 3 mod 4 but the period of continued fractions of √''n is even. 155 is the smallest base for which no generalized Cullen primes or generalized Woodall primes are known. 156 is the largest number n'' such that all primes between ''n/2 and n'' yield a representation as a sum of two primes. 157 is the smallest primorial prime which is not from twin primes. 160 is conjectured to be the only number not of the form ''t+''p'', with t'' triangular number (including 0 and 1), ''p either prime or 0. 161 is conjectured to be the largest number n'' such that σ(''n)−''n'' is odd but there are no k'' ≠ ''n such that σ(k'')−''k = σ(n'')−''n. 163 is the number of space groups, not including handedness. 164 is the smallest number which is a member of amicable pairs. 165 is the smallest n'' such that ''x''2−''ny''2 is not solvable, but ''x''2−''2ny''2 is. 167 is the only number that cannot be written as a sum of 30 or fifth powers. 169 is the smallest number ''n besides 1 and 9 for which σ(φ(n'')) = φ(σ(''n)). 16E is not a quadratic residue mod any number 3≤''n''≤24, besides, 16E is also a primitive root mod any number 1≤''n''≤16 which have a primitive root. 170 is the smallest even number n'' such that the numerator of the ''n''th Bernoulli number is divisible by a nontrivial square number that is relatively prime to ''n. 171 is the number of different projective configurations of type (103103), in which 10 points and 10 lines meet with 3 lines through each of the points and 3 points on each of the lines, all of which may be realized by straight lines in the Euclidean plane. 172 is the number of space groups, including handedness. 173 is the smallest number with ≥3 odd prime factors whose cyclotomic polynomial has all coefficients ±1. 174 is the number of digits of 100!. 175 has a palindromic reciprocal: 0.0074EE470074EE4700... 179 is conjectured to be the smallest Lychael number. 17E is the largest number that cannot be written as a sum of 8 or fewer cubes. 180 is the kissing number in 8 dimensions. (note that the true value of the kissing number is only known in 1, 2, 3, 4, 8, and 20 dimensions) 181 is the smallest (and the only known) 3-Wall-Sun-Sun prime. 182 is the smallest n'' such that ''n, n''+1, ''n+2, and n''+3 have the same number of divisors. 183 is the smallest Frugal number. 186 is the smallest number ''n for which it is known that there is an infinite number of prime gaps no larger than n''. 187 is the largest proper divisor of the smallest Hardy-Ramanujan number (1001). 188 is the smallest number n>1 for which the arithmetic, geometric, and harmonic means of φ(n) and σ(n) are all integers. 18E is the smallest number that can be formed in more than one way by summing three positive cubes. 190 is the 5th central binomial coefficient. 191 is the smallest non-trivial triangular star number. 193 is the smallest perfect totient number to be neither a power of three nor thrice a prime. 194 is the value of 24 (where ''nm'' is the tetration). 195 is the only known Fermat prime which is irregular prime. 198 is the constant of an 8×8 magic square. 199 is the ''n for which the smallest prime of the form n''×10''k−1 is largest for all n'' < 274 (the smallest generalized Riesel number base 10). 19E is the smallest prime ''p such that (p''−1)/2 is irregular prime. (note that 19E itself is also irregular prime) 1X0 is the largest number ''n such that carmichael_lambda(n'') = X. 1X3 is the number of groups with order 26 (=54). 1X5 is the smallest prime base for which no generalized repunit (probable) primes are known. 1X7 is the smallest prime ''p such that neither p''−1 nor ''p+1 is cubefree. 1E1 is the 8th Euler (or up/down) number. 1E4 is the base with the largest conjectured smallest generalized Sierpinski number and the largest conjectured smallest generalized Riesel number in all bases ≤1000 (this two numbers are in order 924XE44391X56 and 49731795912E69). 1E5 is the largest prime p such that (1!+2!+3!+4!+ ... +p!) - 2 is prime. 1E7 is the smallest n'' such that φ7(''n) > 1. 1E8 is the smallest n'' appearing twice in ''P union Q'' union ''R defined with: Construct sequences P'', ''Q, R'' by the rules: ''Q = first differences of P'', ''R = second differences of P'', ''P starts with 1, 3, 9, Q'' starts with 2, 6, ''R starts with 4; at each stage the smallest number not yet present in P'', ''Q, R'' is appended to ''R. 1E0 is the smallest number whose aliquot sequence has not yet been fully determined. 1EX is the smallest nonsemiprime which is a possible value of the smallest (Fermat) prime base n''. 200 is the smallest ''n>8 such that both n'' and ''n+1 are powerful. 202 is the smallest n'' such that a positive definite integral quadratic form is universal if it takes the numbers from 1 to ''n as values. (a more precise version states that, if an integer valued integral quadratic form represents all the numbers 1, 2, 3, 5, 6, 7, X, 11, 12, 13, 15, 17, 19, 1X, 1E, 22, 25, 26, 27, 2X, 2E, 31, 36, 4X, 79, 92, 101, 14E, 202, then it represents all positive integers) 203 is the largest number that is not the sum of distinct non-trivial powers. 204 is the 5th term of the continued fraction of π. 208 is conjectured to be the smallest n'' which is not power of 10 and not congruent to 1 mod 11 (in which all such numbers are divisible by 11) and not congruent to EX mod EE (in which all such numbers are divisible by either E or 11) such that (''nk''.1) (dot means concatenation) cannot be prime. (the only smaller ''n with unknown status are 117, 153 and 172) 210 is the smallest nonsquare number which is not a primitive root mod any safe prime. 214 is the 6th primitive abundant number. 21E is the smallest n'' such that the repunit with length ''n has no known proper factors (except 1). 221 is the number of intersections when all the diagonals of a regular dozagon are drawn. 222 is the smallest happy number which is not power of 10. 223 is the smallest odd number n'' such that φ(''n) < φ(n''−1). 225 is the smallest number ''n such that kn+1 is not prime for all k''≤20. 22E is the smallest Fibonacci U-pseudoprime. 230 is the smallest untouchable number which is a square. 23E is the number of degree 10 irreducible polynomials over GF(2). 241 is the smallest odd number ''n such that |2''k''−''n''| is composite for all 1≤''k''≤''n''. 245 is the smallest (Fermat) pseudoprime base 2 (also called Sarrus number or Poulet number). 247 is the only cube which is a nontrivial repunit in some base (base 16). 255 is the smallest number whose 4th power can be written as the sum of four 4th powers. 260 is the smallest number with ≥20 divisors. 262 is the smallest even base for which no generalized Carol primes are known. 265 is the smallest number that can be written as a sum of consecutive squares in more than 1 way. 269 is the number of octominoes (8-minoes). 271 is conjectured to be the largest prime p'' such that (previous prime(''p))# ± p'' are both primes. 274 is the smallest generalized Riesel number base 10. 275 is the largest Fibonacci number ''n such that the period length of 1/''n'' is ≤10. 276 is the smallest n'' ≠ X mod E for which there are no non-titanic prime of the form ''n×10''k''+1. 277 is the largest known prime p'' such that ''p! − 1 and p''# + 1 are both primes. 278 is the smallest even base not of the form n^x (where generalized repunits can be factored algebraically) for which no generalized repunit (probable) primes are known. 280 is the order of the hyperoctahedral group for ''n = 4. 281 is the smallest integer such that the factorization of x''n''−1 over Q'' includes coefficients other than ±1 and ±2. 2902''n appears in a denominator of an infinite product of π. 293 is the smallest Lucas-Carmichael number. 298 is the n'' for which the smallest prime of the form ''n×10''k''+1 is largest for all n'' < 375 (the smallest generalized Sierpinski number base 10). 2X1 is conjectured to be the largest prime ''p whose smallest primitive root is larger than √''p''. 2XX is the smallest non-primepower k'' such that binomial(2''k, k'') = 2 (mod ''k). (besides, 2XX is also the only known such even k'') 2EE is the smallest prime ''p>E ends with E'' such that the period length of 1/''p is not (p''−1)/2. 309 is the smallest number with more than one factorization into ''S-primes. (Let S'' = {1, 5, 9, 11, 15, 19, ..., 4''k+1, ...}; then an S''-prime is in ''S but is not divisible by any members of S'' except itself and 1. 319 is the number requiring the largest base (342863E) to be the primary pretender. 31X is the largest number that cannot be written as a sum of 7 or fewer cubes. 321 is the smallest prime which is concatenation of numbers from some number (3) down to 1. 326 is conjectured to be the largest base ''b for which there are no (Fermat) pseudoprimes ≤''b''+1. 330 is the largest module for the known property of odd perfect numbers. (the known property of odd perfect numbers is = 1 mod 10, or = 99 mod 330, or = 69 mod 230) 340 is the largest number n'' such that carmichael_lambda(''n) = 8. 344 is the smallest square number which is nontotient. 34E is the smallest irregular prime with irregular index greater than 2. 350 is the smallest number of faces such that holyhedron is known to exist. 353 is the largest sum-product number. 354 is the third perfect number. 358 is the smallest strong Achilles number. 360 is the largest number n'' such that carmichael_lambda(''n) = 6. 368 is the only known cube n'' such that ''n×2''n''−1 is prime (Woodall prime). (note that its cube root is also a cube) 369 is the smallest nonsquare automorphic number. 373 is the number with the lowest property (for random base) to be the primary pretender. 375 is the square root of the smallest Perrin pseudoprime. (note that the smallest Perrin pseudoprime (the square of 375) is 111101, a near-repunit number, and contains only five 1's and one 0, no any digit >1) 378 is the starting number of the first run of 15 consecutive composite numbers. 380 is the smallest number which can not be made prime by changing one of its digits. 388 is the smallest number > e''2π. 3X8 is the smallest number which is a Rhonda number in some base (base 10). 3X9 is the smallest Carmichael number. 3XX is the smallest number not itself an amicable pair which terminates at an amicable pair. 3XE is the largest known Wilson prime. 3E8 is conjectured to be the largest base ''b for which there are no (Fermat) pseudoprimes ≤''b''−1. 404 is the smallest number n'' such that φ(''x) = n'' has only two solutions and the smaller of this two solutions is not prime power (including 1). 420 is the largest possible number of cells of 4-dimention polytope. (note that for ''n-dimention polytope, n''≥5, the only possible number of cells are ''n+1, 2''n'', and 2''n'') 42E is the smallest number whose square is an even-digit palindromic number. 455 is the smallest prime which is a prime factor of a composite Fermat number. 470 is the smallest n'' (and the only ''n≤1000) such that k''×''n is Harshad number for all k''≤600000. (the smallest ''k such that 470''k'' is not Harshad number is 750275) 497 is the first irregular prime to appear in the numerator of a Bernoulli number. 4X5 is the length of a repunit prime. 4X9 is the smallest Fibonacci V-pseudoprime. 4E6 is conjectured to be the largest number n'' such that ''n×(n''+1) is a primorial (15#). 500 is the smallest number which cannot be written as ''mn with both m'' and ''n are powers of squarefree numbers (including 1). 520 is the constant term of modular function j as power series in q=e^(2\pi i t). 545 is the smallest odd number n'' such that 2''k+''n'' is composite for all 1≤''k''≤''n''. 551 is the period of the sequence of Bell numbers mod 5. 598 is the smallest weird number which also an untouchable number. 5X0 is the largest number n'' such that ''k^2 mod n'' is square number for all ''k coprime to n. 5E6 is the Kaprekar constant for 3-digit numbers. 620 is the starting number of the first run of 17 consecutive composite numbers. 635 is the smallest composite de Polignac number. 666 is the "beast number". 668 is the smallest 3-digit narcissistic number. 669 is the smallest odd abundant number. 66X is conjectured to be the largest even number which is (Fermat) pseudoprime to 1/4 of the bases coprime to it. 6X5 is the smallest extra strong Lucas pseudoprime. 6X9 is conjectured to yields the highest residue for 2^h(n)/(3^t(n)*n), where h and t are the number of halving resp. tripling steps in the Collatz problem. 704 is conjectured to be the largest base b'' for which there are no (Fermat) pseudoprimes <''b−1. 771 is the smallest Wieferich prime. 77E is the number of steps for the Conway's game of Life starting with the "F-pentomino" to stabilize. 780 appears in the aliquot sequence of 780/4 (=1E0), which is the smallest number whose aliquot sequence has not yet been fully determined. 781 is the smallest deceptive prime which is not semiprime. 782 is the smallest number n'' which is (Fermat) pseudoprime to exactly 11 bases 0≤''b≤''n''−1. (note that for all 1≤''k''≤11, but not for k''=12, there exists number ''n which is (Fermat) pseudoprime to exactly k'' bases 0≤''b≤''n''−1, and for k''=11, the smallest such number ''n is largest for all these values of k'') 7X2 is the starting number of the first run of 19 consecutive composite numbers. 8X4 is the smallest ''n>7 such that n''! is not Harshad number. 8X7 is an exponent of Mersenne primes. 928 is the starting number of the first run of 29 consecutive composite numbers. (no other such number <5640) X45 is conjectured to be the largest Stern prime. X83 is the only 3-digit narcissistic number with distinct digits. X85 is the smallest ''n such that the number of bases b'' mod ''n for which b''n''−1 = 1 mod n'' is a nontotient (344). X91 is the smallest number which is not the sum of a perfect power (including 1, but not including 0) and a prime. E60 is the smallest even base for which no generalized Carol primes or generalized Kynea primes are known. E80 is the largest number ''n such that k''^2 mod ''n is prime power for all k'' coprime to n. EE6 is conjectured to be the largest number ''n which is not quadratic residue mod all primes p''≤√''n not dividing n''. The famous Hardy-Ramanujan number 1001 1001 is the smallest number which can be written as the sum of of 2 positive cubes in 2 different ways. 1001 is the smallest 4-digit palindromic number. 1001 is the smallest absolute Euler pseudoprime. 1001 is the smallest palindromic number which cannot be prime when read in any base. 1001 is the smallest Carmichael number of the form (6''n+1)×(10''n''+1)×(16''n''+1) with all 6''n''+1, 10''n'+1, and 16''n''+1 primes.'' Beyond 1001 10X0 is conjectured to be the largest number n'' such that −''n is not quadratic residue mod all primes p''≤√''n not dividing n''. (i.e. this number is the largest Euler's "numerus idoneus" (or convenient numbers, or idoneal numbers)) 1129 is the smallest Euler-Jacobi pseudoprime base 2 which is not Carmichael number. 1154 is the smallest number which is value of the Euler totient function but not of the Carmichael lambda function. 11XX is the only non-3-digit ''n with sum of cubes of digits of n'' equals ''n. 1227 is the smallest strong pseudoprime base 2. 1420 is the number of groups with order X8. 1560 is the largest number such that no number less than its twice has more divisors. 1685 is the smallest generalized Wieferich prime base 10. 16E6 is the denominator of the first Bernoulli number which has an irregular pair (B''10 = 497/16E6, which has an irregular pair (497, 10)). 18X3 is conjectured to be the only number to appear 7 or more times in Pascal's triangle. 2467 is the smallest prime with 10 the smallest primitive root. 2497 is the smallest cyclic number. 299E is the smallest prime which is the larger of a Wieferich pair with both primes are odd primes. 2X04 is the only square number besides 0 and 1 which is also a square pyramidal number. 2E00 is the largest factorial number which is also highly composite. 2E01 is the largest square number of the form ''n!+1. 3415 is the smallest odd number which is not of the form 2''n''2+''p'' with p'' prime. 3575 is conjectured to be the largest odd number which is not of the form 2''n''2+''p with p'' prime. 4170 is the largest triangular number which is also a tetrahedral number. 4451 is conjectured to be the largest prime number which is not the sum of another prime number and a nonzero square. 48X7 is conjectured to be the largest number which is a repunit in three or more bases (not including base 1). 5X1E is the smallest base such that there are no primary pretenders < the smallest Carmichael number (3X9). 866E is the only known nontrivial Wieferich-non-Wilson prime. 98E7 is the smallest Wolstenholme prime. E414 is the largest square number which is also a tetrahedral number. E801 is the largest number which is not the sum of two abundant numbers. EEE5 is conjectured to be the largest non-repunit permutable prime. 10667 is conjectured to be the largest number which is not the sum of a prime number and a nonzero square. 11031 is the smallest happy prime. 12345 is the smallest Smarandache prime. 13665 is the smallest prime ''p such that the number of primes end with 1 or 5 ≤ p'' is more than the number of primes end with 3, 7 or E ≤ ''p. (of course, 3 is the only prime ends with 3) 13885 is the smallest non-primepower k'' such that binomial(2''k−1, k''−1) = 1 (mod ''k). 14060 is the smallest number divisible by every natural number from 1 to 10. 14X28 is the smallest number not ends with 0 with is an order of nonsolvable group. 16661 is the smallest palindromic square with a non-palindromic square root. 16E61 is the largest square of the form 2''n''−7 (related to the Ramanujan–Nagell equation). 23000 is conjectured to be the largest number that can be written as a sum of a perfect power (including 1, but not including 0) and a prime in a unique way. 2E69E is the smallest prime p'' such that the number of primes end with 1 or E ≤ ''p is more than the number of primes end with 5 or 7 ≤ p''. 31E00 is the largest number ''n such that carmichael_lambda(n'') = 10. 31E14 is the value of 253 (where ''a[n'']''b is the hyperoperation). 31E15 is conjectured to be the largest Fermat prime. 39565 is conjectured to be the smallest Sierpinski number. 511E7 is the largest two-sided prime. 63X00 is the largest number k'' such that for any positive integers ''x, y'' coprime to ''k, x''x'' = y'' (mod ''k) if and only if y''y'' = x'' (mod ''k). 7923X is the smallest even weak pseudoprime base 2. 95900 is the kissing number in 20 dimensions. X0693 is the largest triangular number which is also a square pyramidal number. 110XX1 is conjectured to be the smallest prime Sierpinski number. 111101 is the smallest Perrin pseudoprime. 160061 is the smallest palindromic square with an even number of digits. 206817 is conjectured to be the smallest Riesel number. 37761E is the smallest Perrin pseudoprime which is not prime power. 562E31 is the smallest number which is a strong pseudoprime to both base 2 and base 3. 715261 is conjectured to be the largest number which is not the sum of a perfect power (including 1, but not including 0) and a prime. E77115 is conjectured to be the largest non-repunit circular prime. 2X3X00X is the smallest even Fibonacci pseudoprime. 5653662 is the smallest even Perrin pseudoprime. 6E8XE77 is the smallest weakly prime. 9321341 is the smallest restricted Perrin pseudoprime. E2X20X8 is the smallest power of 2 starts with E. 24727225 is conjectured to be the only unique prime not of the form Φ''n''(10), where Φ is the cyclotomic polynomial. 2696E854 is the smallest friend of 20. XX000001 is the largest minimal prime. 213574615 is the smallest n'' > 1 such that (−1)Ω(1) + (−1)Ω(2) + ... + (−1)Ω(''n) > 0 (where Ω(k'') is number of primes dividing ''k (counted with multiplicity)). (i.e. this number is the smallest counterexample to Polya's conjecture) 4EE2308X7 is the maximum positive value for a 28-bit signed binary integer in computing. 9EX461593 is conjectured to be the largest odd number which is number of sides of a constructible polygon. 9EX461595 is the smallest composite Fermat number. 375EE5E515 is the largest right-truncatable prime. 9X03693X831 is the smallest prime p'' such that the number of primes end with 1 or 7 ≤ ''p is more than the number of primes end with 2, 5 or E ≤ p (of course, 2 is the only prime ends with 2). 1023456789XE is the smallest pandigital number. 821000001000 is the only one autobiographical number. 606890346850EX6800E036206464 is the largest polydivisible number. 471X34X164259EX16E324XE8X32E7817 is the largest left-truncatable prime. 15079346X6E3E14EE56E395898E96629X8E01515344E4E0714E is the largest narcissistic number. Sequence of uninteresting numbers Numbers that are not (primes, E-smooth, perfect powers, or palindromes): 2X, 32, 3X, 43, 49, 4X, 52, 58, 59, 62, 64, 6X, 71, 72, 73, 78, 79, 7X, 7E, 86, 8X, 93, 96, 97, 98, 9X, 9E, X2, X3, X4, X9, E1, E2, E4, E6, E9, EX, 102, 104, 108, 109, 10E, 110, 112, 113, 115, 118, 11X, 122, 123, 124, 126, 129, 12X, 132, 133, 134, 135, 136, 137, 138, 13X, 142, 143, 149, 14X, 14E, 150, 152, 153, 154, 155, 158, 159, 15X, 15E, 162, 163, 165, 166, 16X, 170, 172, 174, 176, 177, 178, 179, 17X, 184, 186, 187, 188, 189, 192, 193, 196, 197, 198, 199, 19X, 1X2, 1X3, 1X4, 1X8, 1X9, 1XX, 1E0, 1E2, 1E3, 1E6, 1E8, 1E9, 1EX, 1EE, 203, 204, 207, 208, 20X, 20E, 211, 213, 214, 215, 216, 219, 21X, 220, 224, 226, 227, 229, 22X, 22E, 231, 233, 234, 235, 238, 239, 23X, 23E, 243, 244, 245, 246, 248, 249, 24X, 250, 253, 256, 257, 258, 259, 25X, 264, 265, 266, 268, 269, 26X, 26E, 270, 274, 275, 278, 279, 27X, 283, 284, 286, 287, 289, 28X, 28E, 293, 296, 297, 298, 29X, 29E, 2X0, 2X3, 2X4, 2X5, 2X6, 2X7, 2X8, 2X9, 2XX, 2E3, 2E4, 2E5, 2E6, 2E7, 2E8, 2E9, 2EX, 302, 304, 305, 306, 30X, 310, 311, 312, 317, 318, 319, 31X, 31E, 320, 322, 324, 328, 329, 32X, 330, 331, 332, 334, 335, 336, 337, 338, 339, 33X, 341, 342, 345, 348, 349, 350, 351, 352, 354, 355, 356, 359, 35X, 361, 362, 364, 366, 367, 369, 36X, 36E, 370, 371, 372, 374, 376, 378, 37X, 37E, 382, 384, 385, 386, 387, 388, 389, 38X, 392, 394, 395, 396, 398, 399, 39E, 3X0, 3X1, 3X2, 3X4, 3X6, 3X7, 3X9, 3XX, 3E0, 3E1, 3E2, 3E4, 3E6, 3E8, 3E9, 3EX, 3EE, 402, 403, 405, 406, 407, 408, 409, 40X, 411, 412, 413, 417, 418, 419, 41X, 422, 423, 426, 428, 429, 42X, 42E, 430, 432, 433, 436, 438, 439, 43X, 43E, 440, 442, 443, 445, 448, 449, 44X, 44E, 450, 451, 452, 453, 456, 458, 459, 45X, 461, 462, 463, 466, 467, 468, 469, 46X, 472, 473, 475, 476, 477, 478, 479, 47X, 47E, 482, 486, 487, 488, 489, 48X, 490, 491, 493, 495, 496, 498, 49X, 49E, 4X0, 4X1, 4X2, 4X3, 4X6, 4X7, 4X9, 4XX, 4XE, 4E0, 4E2, 4E3, 4E5, 4E6, 4E7, 4E8, 4E9, 4EX, 501, 502, 503, 504, 508, 50X, 50E, 510, 512, 514, 516, 518, 519, 51X, 520, 521, 522, 523, 524, 528, 529, 52X, 52E, 532, 533, 534, 536, 537, 538, 539, 53X, 53E, 543, 544, 546, 547, 548, 549, 54X, 54E, 550, 551, 552, 553, 556, 558, 559, 55X, 55E, 561, 562, 563, 564, 566, 567, 569, 56X, 56E, 570, 571, 572, 573, 574, 578, 579, 57X, 57E, 580, 581, 582, 583, 584, 586, 588, 58X, 590, 592, 593, 594, 596, 597, 598, 599, 59X, 5X2, 5X3, 5X4, 5X6, 5X8, 5X9, 5XX, 5XE, 5E0, 5E2, 5E3, 5E4, 5E6, 5E8, 5E9, 5EX, 601, 602, 603, 604, 605, 607, 608, 609, 60X, 610, 612, 613, 618, 619, 61X, 620, 621, 622, 624, 625, 627, 629, 62X, 62E, 631, 632, 633, 634, 635, 638, 639, 63X, 640, 641, 642, 643, 644, 645, 648, 649, 64X, 64E, 651, 652, 653, 654, 657, 658, 659, 65X, 65E, 660, 662, 663, 664, 667, 668, 66X, 670, 671, 672, 673, 674, 677, 678, 679, 67X, 67E, 682, 683, 684, 685, 689, 68X, 691, 692, 693, 694, 697, 699, 69X, 6X0, 6X1, 6X2, 6X3, 6X4, 6X5, 6X8, 6X9, 6XX, 6XE, 6E0, 6E2, 6E3, 6E5, 6E7, 6E8, 6E9, 6EX, 6EE, 702, 703, 704, 706, 708, 709, 70X, 710, 712, 713, 715, 716, 718, 71X, 720, 722, 723, 724, 725, 726, 728, 729, 72X, 72E, 730, 731, 732, 733, 734, 738, 739, 73X, 73E, 741, 742, 743, 744, 746, 748, 749, 74X, 74E, 750, 752, 753, 754, 755, 756, 758, 759, 75E, 761, 762, 763, 764, 765, 766, 768, 76X, 770, 772, 773, 774, 776, 779, 77X, 780, 781, 782, 783, 784, 786, 788, 789, 78X, 78E, 790, 792, 793, 795, 796, 798, 79X, 79E, 7X0, 7X2, 7X3, 7X4, 7X5, 7X8, 7X9, 7XX, 7XE, 7E0, 7E1, 7E2, 7E3, 7E4, 7E5, 7E6, 7E8, 7E9, 7EX, 802, 805, 806, 807, 809, 80X, 810, 811, 812, 813, 814, 815, 816, 819, 81X, 81E, 821, 822, 823, 824, 826, 827, 829, 82X, 831, 832, 833, 834, 836, 837, 839, 83X, 83E, 842, 843, 844, 845, 846, 847, 849, 84E, 850, 852, 854, 856, 857, 859, 85X, 860, 862, 863, 864, 866, 869, 86X, 86E, 870, 872, 873, 874, 875, 876, 877, 879, 87X, 87E, 880, 883, 884, 885, 886, 887, 889, 88X, 891, 892, 893, 894, 895, 896, 897, 899, 89X, 89E, 8X0, 8X1, 8X2, 8X3, 8X4, 8X6, 8X9, 8XX, 8E0, 8E1, 8E2, 8E3, 8E4, 8E6, 8E9, 8EX, 8EE, 902, 903, 904, 906, 908, 90X, 910, 911, 912, 913, 914, 915, 916, 917, 918, 91X, 922, 924, 925, 926, 928, 92X, 930, 931, 932, 933, 934, 935, 936, 937, 938, 93X, 93E, 941, 942, 943, 944, 945, 947, 948, 94X, 94E, 950, 951, 952, 953, 954, 956, 957, 958, 95X, 960, 962, 963, 966, 968, 96X, 96E, 970, 972, 973, 974, 975, 977, 978, 97X, 97E, 980, 981, 982, 983, 984, 985, 986, 98X, 98E, 990, 991, 992, 993, 996, 997, 998, 99X, 99E, 9X0, 9X1, 9X2, 9X3, 9X4, 9X5, 9X6, 9X8, 9XX, 9E0, 9E2, 9E3, 9E4, 9E6, 9E7, 9E8, 9EX, X01, X02, X03, X05, X06, X08, X09, X12, X13, X14, X15, X18, X19, X1E, X20, X21, X22, X23, X24, X25, X28, X29, X2E, X30, X31, X32, X33, X34, X36, X38, X40, X42, X43, X44, X46, X47, X48, X49, X51, X52, X53, X54, X55, X56, X57, X58, X59, X61, X62, X63, X64, X65, X66, X67, X68, X70, X71, X72, X73, X74, X75, X76, X78, X79, X7E, X81, X82, X83, X85, X86, X88, X89, X8E, X90, X92, X93, X94, X96, X97, X98, X99, XX0, XX1, XX2, XX3, XX4, XX5, XX6, XX9, XE0, XE1, XE2, XE4, XE5, XE6, XE8, XE9, E01, E02, E03, E04, E05, E06, E07, E08, E09, E0X, E10, E12, E13, E16, E17, E18, E19, E1X, E20, E22, E23, E24, E26, E27, E28, E2X, E32, E33, E34, E35, E36, E38, E39, E3X, E40, E41, E42, E43, E44, E46, E47, E48, E49, E4X, E50, E51, E52, E53, E54, E55, E57, E58, E59, E5X, E60, E62, E63, E64, E65, E66, E68, E69, E6X, E70, E72, E73, E74, E75, E76, E77, E78, E79, E7X, E82, E83, E84, E85, E86, E87, E88, E89, E8X, E90, E93, E94, E96, E98, E9X, EX0, EX1, EX2, EX3, EX4, EX6, EX7, EX8, EX9, EXX, EE0, EE1, EE2, EE3, EE4, EE6, EE8, EE9, EEX, ... 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